Error-driven Input Modulation: Solving the Credit Assignment Problem without a Backward Pass
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Abstract (arXiv)
Supervised learning in artificial neural networks typically relies on backpropagation, where the weights are updated based on the error-function gradients and sequentially propagated from the output layer to the input layer. Although this approach has proven effective in a wide domain of applications, it lacks biological plausibility in many regards, including the weight symmetry problem, the dependence of learning on non-local signals, the freezing of neural activity during error propagation, and the update locking problem. Alternative training schemes have been introduced, including sign symmetry, feedback alignment, and direct feedback alignment, but they invariably rely on a backward pass that hinders the possibility of solving all the issues simultaneously. Here, we propose to replace the backward pass with a second forward pass in which the input signal is modulated based on the error of the network. We show that this novel learning rule comprehensively addresses all the above-mentioned issues and can be applied to both fully connected and convolutional models. We test this learning rule on MNIST, CIFAR-10, and CIFAR-100. These results help incorporate biological principles into machine learning.
Key facts. Cited by 63 papers in the map (data through 7 Oct 2026). First posted on arXiv 27 Jan 2022.
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Citing papers per year: 2022: 1; 2023: 13; 2024: 21; 2025: 17; 2026: 11.
63 papers.
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- 2023The Predictive Forward-Forward Algorithmcited by 35
- 2023Brain-inspired Computational Intelligence via Predictive Codingcited by 30
- 2023Forward Learning with Top-Down Feedback: Empirical and Analytical Characterizationcited by 13
- 2023Training Convolutional Neural Networks with the Forward-Forward algorithmcited by 13
- 2024Training Machine Learning models at the Edge: A Surveycited by 11
- 2023The Trifecta: Three simple techniques for training deeper Forward-Forward networkscited by 10
- 2024Momentum Auxiliary Network for Supervised Local Learningcited by 9
- 2023Graph Neural Networks Go Forward-Forwardcited by 8
- 2023Improving equilibrium propagation without weight symmetry through Jacobian homeostasiscited by 8
- 2024FFCL: Forward-Forward Net with Cortical Loops, Training and Inference on Edge Without Backpropagationcited by 6
- 2024Advancing Supervised Local Learning Beyond Classification with Long-term Feature Bankcited by 6
- 2024Beyond Backpropagation: Optimization with Multi-Tangent Forward Gradientscited by 6
38 more
- 2023Extending the Forward Forward Algorithmcited by 5
- 2024On-device Online Learning and Semantic Management of TinyML Systemscited by 5
- 2024LLS: Local Learning Rule for Deep Neural Networks Inspired by Neural Activity Synchronizationcited by 5
- 2024MLAAN: Scaling Supervised Local Learning with Multilaminar Leap Augmented Auxiliary Networkcited by 5
- 2024HPFF: Hierarchical Locally Supervised Learning with Patch Feature Fusioncited by 5
- 2024Counter-Current Learning: A Biologically Plausible Dual Network Approach for Deep Learningcited by 5
- 2024Poor Man's Training on MCUs: A Memory-Efficient Quantized Back-Propagation-Free Approachcited by 5
- 2025Backpropagation-free Spiking Neural Networks with the Forward-Forward Algorithmcited by 5
- 2024Towards Interpretable Deep Local Learning with Successive Gradient Reconciliationcited by 4
- 2025TESS: A Scalable Temporally and Spatially Local Learning Rule for Spiking Neural Networkscited by 4
- 2023Feed-Forward Optimization With Delayed Feedback for Neural Network Trainingcited by 3
- 2024LightFF: Lightweight Inference for Forward-Forward Algorithmcited by 3
- 2024Towards training digitally-tied analog blocks via hybrid gradient computationcited by 3
- 2024Temporal-Difference Learning Using Distributed Error Signalscited by 3
- 2025Mono-Forward: Revisiting Forward-Forward through Objective-Locality Decompositioncited by 3
- 2025MAN++: Scaling Momentum Auxiliary Network for Supervised Local Learning in Vision Taskscited by 3
- 2023Polarity is all you need to learn and transfer fastercited by 2
- 2023Node Perturbation Can Effectively Train Multi-Layer Neural Networkscited by 2
- 2024DFA-GNN: Forward Learning of Graph Neural Networks by Direct Feedback Alignmentcited by 2
- 2025Closed-Form Feedback-Free Learning with Forward Projectioncited by 2
- 2025LANCE: Low Rank Activation Compression for Efficient On-Device Continual Learningcited by 2
- 2026Covariance-Aware Goodness for Scalable Forward-Forward Learningcited by 2
- 2023Understanding Predictive Coding as an Adaptive Trust-Region Methodcited by 1
- 2023Intrinsic Biologically Plausible Adversarial Robustnesscited by 1
- 2024Learning Using a Single Forward Passcited by 1
- 2024Forward Learning of Graph Neural Networkscited by 1
- 2024On the Improvement of Generalization and Stability of Forward-Only Learning via Neural Polarizationcited by 1
- 2024Replacement Learning: Training Vision Tasks with Fewer Learnable Parameterscited by 1
- 2025Stochastic Layer-wise Learning: Scalable and Efficient Alternative to Backpropagationcited by 1
- 2025Adaptive Spatial Goodness Encoding: Advancing and Scaling Forward-Forward Learning Without Backpropagationcited by 1
- 2025Traces Propagation: Memory-Efficient and Scalable Forward-Only Learning in Spiking Neural Networkscited by 1
- 2025Feedback Alignment Meets Low-Rank Manifolds: A Structured Recipe for Local Learningcited by 1
- 2025NetworkFF: Unified Layer Optimization in Forward-Only Neural Networkscited by 1
- 2026HCL-FF: Hierarchical and Contrastive Learning for Forward-Forward Algorithmcited by 1
- 2026Conditioned Direct Feedback Alignment via Activity and Error Geometrycited by 1
- 2022Biologically Plausible Training of Deep Neural Networks Using a Top-down Credit Assignment Networkcited by 0
- 2024Training Large-Scale Optical Neural Networks with Two-Pass Forward Propagationcited by 0
- 2025Forward Target Propagation: A Forward-Only Approach to Global Error Credit Assignment via Local Lossescited by 0
50 most cited of 63 All 63 (JSON)
- 2026SOLO: Pretraining Billion-Parameter Language Models with Shared-Output Local Learning28 Sep 2026
- 2026Conditioned Direct Feedback Alignment via Activity and Error Geometry20 Jul 2026
- 2026Breaking Chains with Trees: Model-Parallel Deep Learning with O(log N) Time Complexity19 Jun 2026
- 2026What Accuracy and Gradient Cosine Miss: Evaluating Feedback Alignment via Scale Stability, Reference Validity, and Depth Utility19 Jun 2026
- 2026Forward-Only Convolutional Neural Networks with Learnable Channel-Class Assignment7 Jun 2026
- 2026Synthetic Benchmarks Overstate Forward-Forward Scaling: Real-Data Limits of Layer-Local Training4 Jun 2026
- 2026FFR: Forward-Forward Learning for Regression2 Jun 2026
- 2026HCL-FF: Hierarchical and Contrastive Learning for Forward-Forward Algorithm24 May 2026
- 2026Replacement Learning: Training Neural Networks with Fewer Parameters19 May 2026
- 2026Covariance-Aware Goodness for Scalable Forward-Forward Learning5 May 2026
- 2026Unsupervised Local Plasticity in a Multi-Frequency VisNet Hierarchy9 Apr 2026
- 2025Forward Only Learning for Orthogonal Neural Networks of any Depth19 Dec 2025
38 more
- 2025NetworkFF: Unified Layer Optimization in Forward-Only Neural Networks19 Dec 2025
- 2025Feedback Alignment Meets Low-Rank Manifolds: A Structured Recipe for Local Learning29 Oct 2025
- 2025Dopamine-driven synaptic credit assignment in neural networks25 Oct 2025
- 2025LANCE: Low Rank Activation Compression for Efficient On-Device Continual Learning25 Sep 2025
- 2025Traces Propagation: Memory-Efficient and Scalable Forward-Only Learning in Spiking Neural Networks16 Sep 2025
- 2025Adaptive Spatial Goodness Encoding: Advancing and Scaling Forward-Forward Learning Without Backpropagation15 Sep 2025
- 2025Neuro-inspired Ensemble-to-Ensemble Communication Primitives for Sparse and Efficient ANNs19 Aug 2025
- 2025MAN++: Scaling Momentum Auxiliary Network for Supervised Local Learning in Vision Tasks22 Jul 2025
- 2025Neural Architecture Search with Mixed Bio-inspired Learning Rules17 Jul 2025
- 2025Energy-Efficient Supervised Learning with a Binary Stochastic Forward-Forward Algorithm9 Jul 2025
- 2025Forward Target Propagation: A Forward-Only Approach to Global Error Credit Assignment via Local Losses20 May 2025
- 2025Stochastic Layer-wise Learning: Scalable and Efficient Alternative to Backpropagation8 May 2025
- 2025Backpropagation-free Spiking Neural Networks with the Forward-Forward Algorithm19 Feb 2025
- 2025TESS: A Scalable Temporally and Spatially Local Learning Rule for Spiking Neural Networks3 Feb 2025
- 2025Closed-Form Feedback-Free Learning with Forward Projection27 Jan 2025
- 2025Mono-Forward: Revisiting Forward-Forward through Objective-Locality Decomposition16 Jan 2025
- 2024Poor Man's Training on MCUs: A Memory-Efficient Quantized Back-Propagation-Free Approach7 Nov 2024
- 2024Temporal-Difference Learning Using Distributed Error Signals6 Nov 2024
- 2024Beyond Backpropagation: Optimization with Multi-Tangent Forward Gradients23 Oct 2024
- 2024Replacement Learning: Training Vision Tasks with Fewer Learnable Parameters2 Oct 2024
- 2024Counter-Current Learning: A Biologically Plausible Dual Network Approach for Deep Learning30 Sep 2024
- 2024Towards training digitally-tied analog blocks via hybrid gradient computation5 Sep 2024
- 2024On the Improvement of Generalization and Stability of Forward-Only Learning via Neural Polarization17 Aug 2024
- 2024Training Large-Scale Optical Neural Networks with Two-Pass Forward Propagation15 Aug 2024
- 2024HPFF: Hierarchical Locally Supervised Learning with Patch Feature Fusion8 Jul 2024
- 2024Momentum Auxiliary Network for Supervised Local Learning8 Jul 2024
- 2024MLAAN: Scaling Supervised Local Learning with Multilaminar Leap Augmented Auxiliary Network24 Jun 2024
- 2024Towards Interpretable Deep Local Learning with Successive Gradient Reconciliation7 Jun 2024
- 2024DFA-GNN: Forward Learning of Graph Neural Networks by Direct Feedback Alignment4 Jun 2024
- 2024Advancing Supervised Local Learning Beyond Classification with Long-term Feature Bank1 Jun 2024
- 2024LLS: Local Learning Rule for Deep Neural Networks Inspired by Neural Activity Synchronization24 May 2024
- 2024FFCL: Forward-Forward Net with Cortical Loops, Training and Inference on Edge Without Backpropagation21 May 2024
- 2024On-device Online Learning and Semantic Management of TinyML Systems13 May 2024
- 2024LightFF: Lightweight Inference for Forward-Forward Algorithm8 Apr 2024
- 2024Forward Learning of Graph Neural Networks16 Mar 2024
- 2024Training Machine Learning models at the Edge: A Survey5 Mar 2024
- 2024Learning Using a Single Forward Pass15 Feb 2024
- 2023Training Convolutional Neural Networks with the Forward-Forward algorithm22 Dec 2023
50 newest of 63 All 63 (JSON)
- 2023The Predictive Forward-Forward Algorithmcited by 35
- 2023Brain-inspired Computational Intelligence via Predictive Codingcited by 30
- 2023Forward Learning with Top-Down Feedback: Empirical and Analytical Characterizationcited by 13
- 2023Training Convolutional Neural Networks with the Forward-Forward algorithmcited by 13
- 2024Training Machine Learning models at the Edge: A Surveycited by 11
- 2023The Trifecta: Three simple techniques for training deeper Forward-Forward networkscited by 10
- 2024Momentum Auxiliary Network for Supervised Local Learningcited by 9
- 2023Graph Neural Networks Go Forward-Forwardcited by 8
- 2023Improving equilibrium propagation without weight symmetry through Jacobian homeostasiscited by 8
- 2024FFCL: Forward-Forward Net with Cortical Loops, Training and Inference on Edge Without Backpropagationcited by 6
- 2024Advancing Supervised Local Learning Beyond Classification with Long-term Feature Bankcited by 6
- 2024Beyond Backpropagation: Optimization with Multi-Tangent Forward Gradientscited by 6
12 most cited of 63 All 63 (JSON)
- 2026SOLO: Pretraining Billion-Parameter Language Models with Shared-Output Local Learning28 Sep 2026
- 2026Conditioned Direct Feedback Alignment via Activity and Error Geometry20 Jul 2026
- 2026Breaking Chains with Trees: Model-Parallel Deep Learning with O(log N) Time Complexity19 Jun 2026
- 2026What Accuracy and Gradient Cosine Miss: Evaluating Feedback Alignment via Scale Stability, Reference Validity, and Depth Utility19 Jun 2026
- 2026Forward-Only Convolutional Neural Networks with Learnable Channel-Class Assignment7 Jun 2026
- 2026Synthetic Benchmarks Overstate Forward-Forward Scaling: Real-Data Limits of Layer-Local Training4 Jun 2026
- 2026FFR: Forward-Forward Learning for Regression2 Jun 2026
- 2026HCL-FF: Hierarchical and Contrastive Learning for Forward-Forward Algorithm24 May 2026
- 2026Replacement Learning: Training Neural Networks with Fewer Parameters19 May 2026
- 2026Covariance-Aware Goodness for Scalable Forward-Forward Learning5 May 2026
- 2026Unsupervised Local Plasticity in a Multi-Frequency VisNet Hierarchy9 Apr 2026
- 2025Forward Only Learning for Orthogonal Neural Networks of any Depth19 Dec 2025
12 newest of 63 All 63 (JSON)